12 research outputs found
ATP and Presentation Service for Mizar Formalizations
This paper describes the Automated Reasoning for Mizar (MizAR) service, which
integrates several automated reasoning, artificial intelligence, and
presentation tools with Mizar and its authoring environment. The service
provides ATP assistance to Mizar authors in finding and explaining proofs, and
offers generation of Mizar problems as challenges to ATP systems. The service
is based on a sound translation from the Mizar language to that of first-order
ATP systems, and relies on the recent progress in application of ATP systems in
large theories containing tens of thousands of available facts. We present the
main features of MizAR services, followed by an account of initial experiments
in finding proofs with the ATP assistance. Our initial experience indicates
that the tool offers substantial help in exploring the Mizar library and in
preparing new Mizar articles
Premise Selection for Mathematics by Corpus Analysis and Kernel Methods
Smart premise selection is essential when using automated reasoning as a tool
for large-theory formal proof development. A good method for premise selection
in complex mathematical libraries is the application of machine learning to
large corpora of proofs. This work develops learning-based premise selection in
two ways. First, a newly available minimal dependency analysis of existing
high-level formal mathematical proofs is used to build a large knowledge base
of proof dependencies, providing precise data for ATP-based re-verification and
for training premise selection algorithms. Second, a new machine learning
algorithm for premise selection based on kernel methods is proposed and
implemented. To evaluate the impact of both techniques, a benchmark consisting
of 2078 large-theory mathematical problems is constructed,extending the older
MPTP Challenge benchmark. The combined effect of the techniques results in a
50% improvement on the benchmark over the Vampire/SInE state-of-the-art system
for automated reasoning in large theories.Comment: 26 page
Learning-Assisted Automated Reasoning with Flyspeck
The considerable mathematical knowledge encoded by the Flyspeck project is
combined with external automated theorem provers (ATPs) and machine-learning
premise selection methods trained on the proofs, producing an AI system capable
of answering a wide range of mathematical queries automatically. The
performance of this architecture is evaluated in a bootstrapping scenario
emulating the development of Flyspeck from axioms to the last theorem, each
time using only the previous theorems and proofs. It is shown that 39% of the
14185 theorems could be proved in a push-button mode (without any high-level
advice and user interaction) in 30 seconds of real time on a fourteen-CPU
workstation. The necessary work involves: (i) an implementation of sound
translations of the HOL Light logic to ATP formalisms: untyped first-order,
polymorphic typed first-order, and typed higher-order, (ii) export of the
dependency information from HOL Light and ATP proofs for the machine learners,
and (iii) choice of suitable representations and methods for learning from
previous proofs, and their integration as advisors with HOL Light. This work is
described and discussed here, and an initial analysis of the body of proofs
that were found fully automatically is provided
A Deep Reinforcement Learning Approach to First-Order Logic Theorem Proving
Automated theorem provers have traditionally relied on manually tuned
heuristics to guide how they perform proof search. Deep reinforcement learning
has been proposed as a way to obviate the need for such heuristics, however,
its deployment in automated theorem proving remains a challenge. In this paper
we introduce TRAIL, a system that applies deep reinforcement learning to
saturation-based theorem proving. TRAIL leverages (a) a novel neural
representation of the state of a theorem prover and (b) a novel
characterization of the inference selection process in terms of an
attention-based action policy. We show through systematic analysis that these
mechanisms allow TRAIL to significantly outperform previous
reinforcement-learning-based theorem provers on two benchmark datasets for
first-order logic automated theorem proving (proving around 15% more theorems)
Machine Learning Meets The Herbrand Universe
The appearance of strong CDCL-based propositional (SAT) solvers has greatly
advanced several areas of automated reasoning (AR). One of the directions in AR
is thus to apply SAT solvers to expressive formalisms such as first-order
logic, for which large corpora of general mathematical problems exist today.
This is possible due to Herbrand's theorem, which allows reduction of
first-order problems to propositional problems by instantiation. The core
challenge is choosing the right instances from the typically infinite Herbrand
universe. In this work, we develop the first machine learning system targeting
this task, addressing its combinatorial and invariance properties. In
particular, we develop a GNN2RNN architecture based on an invariant graph
neural network (GNN) that learns from problems and their solutions
independently of symbol names (addressing the abundance of skolems), combined
with a recurrent neural network (RNN) that proposes for each clause its
instantiations. The architecture is then trained on a corpus of mathematical
problems and their instantiation-based proofs, and its performance is evaluated
in several ways. We show that the trained system achieves high accuracy in
predicting the right instances, and that it is capable of solving many problems
by educated guessing when combined with a ground solver. To our knowledge, this
is the first convincing use of machine learning in synthesizing relevant
elements from arbitrary Herbrand universes.Comment: 8 pages, 10 figure
Learning to Prove Theorems via Interacting with Proof Assistants
Humans prove theorems by relying on substantial high-level reasoning and
problem-specific insights. Proof assistants offer a formalism that resembles
human mathematical reasoning, representing theorems in higher-order logic and
proofs as high-level tactics. However, human experts have to construct proofs
manually by entering tactics into the proof assistant. In this paper, we study
the problem of using machine learning to automate the interaction with proof
assistants. We construct CoqGym, a large-scale dataset and learning environment
containing 71K human-written proofs from 123 projects developed with the Coq
proof assistant. We develop ASTactic, a deep learning-based model that
generates tactics as programs in the form of abstract syntax trees (ASTs).
Experiments show that ASTactic trained on CoqGym can generate effective tactics
and can be used to prove new theorems not previously provable by automated
methods. Code is available at https://github.com/princeton-vl/CoqGym.Comment: Accepted to ICML 201
Proceedings of the Deduktionstreffen 2019
The annual meeting Deduktionstreffen is the prime activity of the Special Interest Group on Deduction Systems (FG DedSys) of the AI Section of the German Society for Informatics (GI-FBKI). It is a meeting with a familiar, friendly atmosphere, where everyone interested in deduction can report on their work in an informal setting